Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(1030089025\)\(\medspace = 5^{2} \cdot 7^{4} \cdot 131^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.1967079625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.1967079625.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{3} + 6x + 65 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 58 a^{2} + 12 a + 59 + \left(42 a + 20\right)\cdot 67 + \left(19 a^{2} + 22 a + 46\right)\cdot 67^{2} + \left(34 a^{2} + 25 a + 63\right)\cdot 67^{3} + \left(20 a^{2} + 23 a + 10\right)\cdot 67^{4} + \left(15 a^{2} + 64 a + 32\right)\cdot 67^{5} + \left(45 a^{2} + 62 a + 39\right)\cdot 67^{6} + \left(58 a^{2} + 20 a + 14\right)\cdot 67^{7} + \left(59 a^{2} + 28 a + 3\right)\cdot 67^{8} + \left(37 a^{2} + 44 a + 9\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a^{2} + a + 38 + \left(18 a^{2} + 21 a + 54\right)\cdot 67 + \left(59 a^{2} + 12 a + 19\right)\cdot 67^{2} + \left(56 a^{2} + 28 a + 34\right)\cdot 67^{3} + \left(47 a^{2} + 36 a + 2\right)\cdot 67^{4} + \left(55 a^{2} + 66 a + 40\right)\cdot 67^{5} + \left(9 a^{2} + 55 a + 22\right)\cdot 67^{6} + \left(24 a^{2} + 10 a + 62\right)\cdot 67^{7} + \left(5 a^{2} + 41 a + 22\right)\cdot 67^{8} + \left(47 a^{2} + 43 a + 47\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 a^{2} + 26 a + 4 + \left(25 a^{2} + 37 a + 54\right)\cdot 67 + \left(36 a^{2} + 43 a + 48\right)\cdot 67^{2} + \left(39 a^{2} + 57 a + 17\right)\cdot 67^{3} + \left(39 a^{2} + 30 a + 20\right)\cdot 67^{4} + \left(29 a^{2} + 12 a + 22\right)\cdot 67^{5} + \left(14 a^{2} + 36 a + 50\right)\cdot 67^{6} + \left(42 a^{2} + 60 a + 15\right)\cdot 67^{7} + \left(16 a^{2} + 31\right)\cdot 67^{8} + \left(4 a^{2} + 4 a + 8\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a^{2} + 29 a + 21 + \left(40 a^{2} + 54 a + 44\right)\cdot 67 + \left(11 a^{2} + 16\right)\cdot 67^{2} + \left(60 a^{2} + 51 a + 33\right)\cdot 67^{3} + \left(6 a^{2} + 12 a + 23\right)\cdot 67^{4} + \left(22 a^{2} + 57 a + 59\right)\cdot 67^{5} + \left(7 a^{2} + 34 a + 21\right)\cdot 67^{6} + \left(33 a^{2} + 52 a + 46\right)\cdot 67^{7} + \left(57 a^{2} + 37 a + 60\right)\cdot 67^{8} + \left(24 a^{2} + 18 a + 23\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a^{2} + 43 a + 25 + \left(20 a^{2} + 29 a + 59\right)\cdot 67 + \left(55 a^{2} + 52 a + 43\right)\cdot 67^{2} + \left(14 a^{2} + 9 a + 35\right)\cdot 67^{3} + \left(52 a^{2} + 5 a + 44\right)\cdot 67^{4} + \left(61 a^{2} + 38 a + 34\right)\cdot 67^{5} + \left(46 a^{2} + 59 a + 55\right)\cdot 67^{6} + \left(22 a^{2} + 26 a + 54\right)\cdot 67^{7} + \left(48 a^{2} + 33 a + 3\right)\cdot 67^{8} + \left(44 a^{2} + 38\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a^{2} + 46 a + 14 + \left(64 a^{2} + 28 a + 35\right)\cdot 67 + \left(54 a^{2} + 34 a + 42\right)\cdot 67^{2} + \left(36 a^{2} + 61 a + 56\right)\cdot 67^{3} + \left(19 a^{2} + 60 a + 47\right)\cdot 67^{4} + \left(24 a^{2} + 7 a + 18\right)\cdot 67^{5} + \left(7 a^{2} + 17 a + 31\right)\cdot 67^{6} + \left(46 a^{2} + 3 a + 14\right)\cdot 67^{7} + \left(29 a^{2} + 9 a + 63\right)\cdot 67^{8} + \left(56 a^{2} + 15 a + 17\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 45 a^{2} + 45 a + 55 + \left(49 a^{2} + 8 a + 43\right)\cdot 67 + \left(23 a^{2} + 47 a + 51\right)\cdot 67^{2} + \left(15 a^{2} + 62 a + 37\right)\cdot 67^{3} + \left(62 a^{2} + 17\right)\cdot 67^{4} + \left(47 a^{2} + 21 a + 46\right)\cdot 67^{5} + \left(12 a^{2} + 57 a + 52\right)\cdot 67^{6} + \left(65 a^{2} + 36 a + 23\right)\cdot 67^{7} + \left(55 a^{2} + 24 a + 34\right)\cdot 67^{8} + \left(32 a^{2} + 51 a + 57\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 48 a^{2} + 6 a + \left(51 a^{2} + 64 a + 54\right)\cdot 67 + \left(43 a^{2} + 63 a + 24\right)\cdot 67^{2} + \left(7 a^{2} + 24 a + 38\right)\cdot 67^{3} + \left(44 a^{2} + 9 a + 54\right)\cdot 67^{4} + \left(3 a^{2} + 50 a + 32\right)\cdot 67^{5} + \left(2 a^{2} + 13 a + 58\right)\cdot 67^{6} + \left(18 a^{2} + 64 a + 37\right)\cdot 67^{7} + \left(47 a^{2} + 47 a + 56\right)\cdot 67^{8} + \left(26 a^{2} + 52 a + 32\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 62 a^{2} + 60 a + 56 + \left(63 a^{2} + 48 a + 35\right)\cdot 67 + \left(30 a^{2} + 57 a + 40\right)\cdot 67^{2} + \left(2 a^{2} + 13 a + 17\right)\cdot 67^{3} + \left(42 a^{2} + 21 a + 46\right)\cdot 67^{4} + \left(7 a^{2} + 17 a + 48\right)\cdot 67^{5} + \left(55 a^{2} + 64 a + 2\right)\cdot 67^{6} + \left(24 a^{2} + 58 a + 65\right)\cdot 67^{7} + \left(14 a^{2} + 44 a + 58\right)\cdot 67^{8} + \left(60 a^{2} + 37 a + 32\right)\cdot 67^{9} +O(67^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(4,7)$ | $0$ |
| $18$ | $2$ | $(1,4)(2,7)(6,8)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,5)(4,7)$ | $0$ |
| $27$ | $2$ | $(3,5)(4,7)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,5)(4,7)(6,9)$ | $0$ |
| $6$ | $3$ | $(3,5,9)$ | $-4$ |
| $8$ | $3$ | $(1,6,2)(3,9,5)(4,8,7)$ | $-1$ |
| $12$ | $3$ | $(1,2,6)(3,5,9)$ | $2$ |
| $72$ | $3$ | $(1,3,4)(2,5,7)(6,9,8)$ | $2$ |
| $54$ | $4$ | $(3,7,5,4)(8,9)$ | $0$ |
| $162$ | $4$ | $(2,6)(3,7,5,4)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,7)(3,5,9)(6,8)$ | $-2$ |
| $36$ | $6$ | $(3,8,9,7,5,4)$ | $-2$ |
| $36$ | $6$ | $(3,5,9)(4,7)$ | $0$ |
| $36$ | $6$ | $(1,2,6)(3,5,9)(4,7)$ | $0$ |
| $54$ | $6$ | $(1,6,2)(3,5)(4,7)$ | $0$ |
| $72$ | $6$ | $(1,7,2,8,6,4)(3,5,9)$ | $1$ |
| $108$ | $6$ | $(1,3,2,5,6,9)(4,7)$ | $0$ |
| $216$ | $6$ | $(1,3,7,2,5,4)(6,9,8)$ | $0$ |
| $144$ | $9$ | $(1,3,8,6,9,7,2,5,4)$ | $-1$ |
| $108$ | $12$ | $(1,2,6)(3,7,5,4)(8,9)$ | $0$ |